Python API: Custom properties for vertices/edges/faces?
Only objects which are derived from bpy.types.ID support custom properties. These subclasses are listed here: API docs - bpy.types.ID. You access them via bpy.data.

Additionally, Bone and PoseBone objects support custom properties as well: API docs - bpy.props.

The standard API does not support custom data for geometry elements (bpy).The BMesh Module  (bmesh) does however.

Note that custom data layers are rather slow when utilized with Python.

Keep in mind the "propagation" of custom data, the attributes are copied on operations like extrusion - hence, you can't really use it to give each vert / edge / face a unique number for instance.

See also:


Python: assign custom tag to vertices
API Docs - Custom data access